On the problem of classification of finite groups associated to multiplicative $\eta$-products

In this article, we study finite groups such that the cusp forms associated to all elements of these groups by means of some faithful representation are modular forms with multiplicative Fourier coefficients from a special class. The Sylow subgroups of such groups of odd order are found. We consider such metacyclic groups. The groups of order 16 and the groups of order 32 that are metacyclic or are direct products of a group of order 16 and the cyclic group of order 2 are considered in detail.